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E-raamat: Measurement and Probability: A Probabilistic Theory of Measurement with Applications

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Measurement and Probability: A Probabilistic Theory of Measurement with ApplicationsSuurem pilt
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Measurement plays a fundamental role both in physical and behavioral sciences, as well as in engineering and technology: it is the link between abstract models and empirical reality and is a privileged method of gathering information from the real world.

Is it possible to develop a single theory of measurement for the various domains of science and technology in which measurement is involved? This book takes the challenge by addressing the following main issues: What is the meaning of measurement? How do we measure? What can be measured?

A theoretical framework that could truly be shared by scientists in different fields, ranging from physics and engineering to psychology is developed. The future in fact will require greater collaboration between science and technology and between different sciences. Measurement, which played a key role in the birth of modern science, can act as an essential interdisciplinary tool and language for this new scenario.

A sound theoretical basis for addressing key problems in measurement is provided. These include perceptual measurement, the evaluation of uncertainty, the evaluation of inter-comparisons, the analysis of risks in decision-making and the characterization of dynamical measurement. Currently, increasing attention is paid to these issues due to their scientific, technical, economic and social impact. The book proposes a unified probabilistic approach to them which may allow more rational and effective solutions to be reached.

Great care was taken to make the text as accessible as possible in several ways. Firstly, by giving preference to as interdisciplinary a terminology as possible; secondly, by carefully defining and discussing all key terms. This ensures that a wide readership, including people from different mathematical backgrounds and different understandings of measurement can all benefit from this work. Concerning mathematics, all the main results are preceded by intuitive discussions andillustrated by simple examples. Moreover, precise proofs are always included in order to enable the more demanding readers to make conscious and creative use of these ideas, and also to develop new ones.

The book demonstrates that measurement, which is commonly understood to be a merely experimental matter, poses theoretical questions which are no less challenging than those arising in other, apparently more theoretical, disciplines.

Arvustused

The book develops a theoretical framework that could be shared by scientists in different fields, including physics, engineering, psychology. The book will be useful for those working in the areas of measurability and probability and their applications. (Anatoliy Swishchuk, zbMATH 1391.60002, 2018)







The book is addressed to specialists who deal with measurements in engineering work, metrology and scientific research. The chapter with some applications to acoustical measurements might be of special interest for acousticians and for those who calibrate acoustical instrumentation. Through the whole content, with numerous references in each chapter, the book of Giovanni Battista Rossi is a very valuable text, recommended to all those who are connected to the theory and practice of measurements. (Valentin Buzduga, Noise Control Engineering Journal, Vol. 62, December, 2014)

Part I General Problems
1 Measurability
3(20)
1.1 What Can Be Measured?
3(1)
1.2 Counting and Measuring
4(3)
1.3 Physical Measurement
7(3)
1.4 Psychophysical Measurement
10(2)
1.5 The Debate at the British Association for the Advancement of Science
12(1)
1.6 A Turning Point: Stevens's Twofold Contribution
13(4)
1.6.1 Direct Measurement of Percepts
13(1)
1.6.2 Classification of Measurement Scales
14(3)
1.7 The Representational Theory
17(1)
1.8 The Role of the Measuring System
18(1)
1.9 The Proposed Approach
19(4)
References
21(2)
2 Uncertainty
23(22)
2.1 Why are Measurement Results not Certain?
23(1)
2.2 Historical Background
24(12)
2.2.1 Gauss, Laplace and the Early Theory of Errors
24(2)
2.2.2 Fechner and Thurstone: The Uncertainty of Observed Relations
26(4)
2.2.3 Campbell: Errors of Consistency and Errors of Methods
30(1)
2.2.4 The Contribution of Orthodox Statistics
31(1)
2.2.5 Uncertainty Relations in Quantum Mechanics
32(2)
2.2.6 The Debate on Uncertainty at the End of the Twentieth Century
34(2)
2.3 The Proposed Approach
36(9)
2.3.1 Uncertainty Related to the Measurement Scale and to Empirical Relations
37(2)
2.3.2 Uncertainty Related to the Measurement Process and the Measuring System
39(1)
2.3.3 Information Flux Between the Objects(s) and the Observer
40(1)
References
41(4)
Part II The Theory
3 The Measurement Scale: Deterministic Framework
45(48)
3.1 What is the Meaning of Measurement?
45(2)
3.2 The General Framework
47(2)
3.2.1 Overview
47(1)
3.2.2 Some Formal Statements
48(1)
3.2.3 Overview of the Main Types of Scales
49(1)
3.3 Ordinal Scales
49(6)
3.3.1 Motivations for Dealing with Ordinal Scales
49(1)
3.3.2 Serialising and Numbering Objects
50(2)
3.3.3 Representation for Order Structures
52(3)
3.4 Interval Scales
55(10)
3.4.1 Dealing with Intervals
55(1)
3.4.2 Measuring Differences
56(3)
3.4.3 Representation for Difference Structures
59(6)
3.5 Ratio Scales for Intensive Structures
65(8)
3.5.1 Is Empirical Addition Necessary for Establishing a Ratio Scale?
65(1)
3.5.2 Extensive and Intensive Quantities
66(1)
3.5.3 Scaling Intensities
67(3)
3.5.4 Representation for Intensive Structures
70(3)
3.6 Ratio Scales for Extensive Structures
73(7)
3.6.1 The Role of Additivity in Measurement
73(3)
3.6.2 Representation for Extensive Structures
76(4)
3.7 Derived Scales
80(8)
3.7.1 Derived Versus Fundamental Scales
80(1)
3.7.2 Representation for Derived Scales
81(4)
3.7.3 Systems of Quantities
85(1)
3.7.4 The International System of Metrology
86(2)
3.8 Summary
88(5)
References
90(3)
4 The Measurement Scale: Probabilistic Approach
93(24)
4.1 Working with Probability
93(14)
4.1.1 The Nature of Probability
93(1)
4.1.2 The Rules of Probability
94(3)
4.1.3 An Illustrative Example
97(3)
4.1.4 Probability as a Logic
100(1)
4.1.5 Probabilistic Variables
100(2)
4.1.6 Probabilistic Functions
102(1)
4.1.7 Probabilistic Relations
103(2)
4.1.8 Continuity
105(1)
4.1.9 Non-probabilistic Approaches to Measurement Uncertainty
106(1)
4.2 Probabilistic Representations
107(1)
4.3 Probabilistic Fundamental Scales
108(3)
4.3.1 Order Structures
108(2)
4.3.2 Difference Structures
110(1)
4.3.3 Intensive Structures
110(1)
4.3.4 Extensive Structures
111(1)
4.4 Probabilistic Derived Scales
111(4)
4.4.1 An Introductory Example
111(2)
4.4.2 Probabilistic Cross-Order Structures
113(2)
4.4.3 Probabilistic Cross-Difference Structures
115(1)
4.5 Summary
115(2)
References
116(1)
5 The Measurement Process
117(30)
5.1 How Can We Measure?
117(4)
5.2 Deterministic Model of the Measurement Process
121(3)
5.3 Probabilistic Model of the Measurement Process
124(3)
5.4 Probability Space of the Measurement Process
127(10)
5.4.1 From Numbers to Numbers
128(4)
5.4.2 From Things to Numbers
132(5)
5.5 Systematic Effects
137(5)
5.6 Continuous Versus Discrete Representations
142(2)
5.7 Overall Probabilistic Framework and Generalisations
144(3)
References
145(2)
6 Inference in Measurement
147(16)
6.1 How Can We Learn from Data?
147(1)
6.2 Probabilistic Models and Inferences
148(7)
6.2.1 The Bernoullian Model
148(1)
6.2.2 A Classification of Probabilistic Inferences
149(6)
6.3 Measurement Evaluation
155(5)
6.4 Measurement Verification
160(1)
6.5 Summary
161(2)
References
162(1)
7 Multidimensional Measurement
163(16)
7.1 What Happens when Moving from One to Two Dimensions
163(2)
7.2 Distances and Metrics
165(2)
7.3 Nominal and Distance Structures
167(5)
7.3.1 Nominal Structures
167(1)
7.3.2 Distance Structures
168(4)
7.4 Probabilistic Representation for Nominal and Metric Structures
172(2)
7.5 Additional Notes on Multidimensional Measurement
174(5)
References
175(4)
Part III Applications
8 Perceptual Measurement
179(26)
8.1 Measuring the Impossible
179(2)
8.2 Measuring the Intensity of a Sensation
181(18)
8.2.1 Premise: Some Acoustic Quantities
181(1)
8.2.2 Loudness of Pure Tones
182(6)
8.2.3 Loudness of Pink Noise
188(2)
8.2.4 Direct Measurement of Loudness: Master Scaling
190(3)
8.2.5 Direct Measurement of Loudness: Robust Magnitude Estimation
193(5)
8.2.6 Indirect Measurement: Loudness Model
198(1)
8.3 State of the Art, Perspective and Challenges
199(6)
References
203(2)
9 The Evaluation of Measurement Uncertainty
205(18)
9.1 How to Develop a Mathematical Model of the Measurement Process
205(12)
9.1.1 Statement of the Problem
205(1)
9.1.2 Linear Models
206(1)
9.1.3 Systematic Effects and Random Variations
207(2)
9.1.4 Observability
209(1)
9.1.5 Low-Resolution Measurement
210(3)
9.1.6 Practical Guidelines
213(1)
9.1.7 Hysteresis Phenomena
214(2)
9.1.8 Indirect Measurement
216(1)
9.2 Measurement Software
217(2)
9.3 A Working Example
219(4)
References
221(2)
10 Inter-Comparisons and Calibration
223(14)
10.1 A Worldwide Quality Assurance System for Measurement
223(1)
10.2 A Probabilistic Framework for Comparisons
224(8)
10.2.1 How Key Comparisons Work
224(1)
10.2.2 Checking the Individual Results
224(2)
10.2.3 The Paradigm of the Probabilistic Scale
226(4)
10.2.4 Summary of the Proposed Approach
230(1)
10.2.5 A Working Example
231(1)
10.3 Calibration
232(5)
References
236(1)
11 Measurement-Based Decisions
237(16)
11.1 The Inferential Process in Conformance Assessment
237(1)
11.2 A Probabilistic Framework for Risk Analysis
238(8)
11.2.1 Insight into Conformance Assessment
238(4)
11.2.2 Probabilistic Framework
242(1)
11.2.3 Illustrative Example
243(3)
11.3 Software for Risk Analysis
246(1)
11.4 Chemical Analysis
246(2)
11.5 Legal Metrology
248(1)
11.6 A Working Example
249(4)
References
251(2)
12 Dynamic Measurement
253(20)
12.1 Dynamic Measurement: An Introduction
253(1)
12.2 Direct Dynamic Measurement
254(11)
12.2.1 A Probabilistic Framework for Direct Dynamic Measurement
254(7)
12.2.2 Evaluation of the Uncertainty Generated by Dynamic Effects in Instrumentation
261(4)
12.3 Indirect Dynamic Measurement: Spectrum Measurement
265(8)
References
271(2)
Appendix A Glossary and Notation 273(8)
Index 281
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